Second case when (REF )-() can be trusted is in holographic theories, i.e. large \(c\)  theories satisfying HKS sparseness condition [1]}. From the holographic point of view \(f_0\)  is the free energy of BTZ black hole in the Euclidean classical theory of gravity with the deformed boundary conditions such that the dual CFT Hamiltonian is \(H=\sum _n \mu _{2n-1} Q_{2n-1}\)  [2]}, [3]}, [4]}. The leading correction \(f_1\)  can be interpreted as the one-loop contribution coming from the boundary gravitons. Different solutions of () means Euclidean path integral could have numerous BTZ saddles and the condition \(h(t_n)>1/12\)  necessary for the validity of Cardy formula would come automatically as the requirement of smoothness of bulk geometry.
